R. L. Plackett and J. P. Burman published their study in 1946. They explain how to build plans to study a large number of factors in few trials. Their hypothesis is that all the interactions between factors are negligible in relation to the effects of the factors. In other words, this type of plan only allows to study the effects of the main factors and not the interactions. This can be seen as fractional factorial plans of resolution III.
Construction of the matrix of experiments and effects
Matrix in the case of this type of matrix, the matrixes of experiments and effects of Plackett-Burman are other than Hadamard matrices1. Plackett and Burman have simplified and systematized their construction2.
We note for the following that n is the number of factor and NH, the number of experiments.
1. Indicate the signs of the first column
The first column of the matrix is constructed, corresponding to the effect of factor 1. We will build it according to the following table:
We notice, that we have a choice of 7 combinations more and less. The combination we are going to choose depends on the number of factors we have to study. Knowing that a table of Plackett and Burman allows to study N-1 factors, the choice is as follows:
- for 1 to 3 factors: NH = 4
- For 4 to 7 factors: NH = 8
- for 8 to 11 factors: NH = 12
For example, if you want to study 7 factors, you will have to do 8 tests whose first column will be + + +-+–.
2. Deduction of N-2 columns
The N-2 other columns are then deducted by circular permutation according to the following diagram:
The previous step is reproduced to obtain a matrix of this type for 8 trials and 7 factors:
3. Add a column and a row
We’ve seen it since step 1, we’re missing a line and a sign column for the last Test. We add them and give them the sign – to get the following matrix:
4. Swap Signs
The objective from this step is to find a standard matrix of complete factorial planes. To do that, you start by numbering the lines. We know that by changing all the signs of a table, we do not change the quality of the results. So we permutons all the signs to get the following result:
5. Order the Lines
Now order the lines to obtain a standard matrix (in our case a 23) full factorial plans. In the end we get this matrix:
We obtain a matrix or factors 4, 5, 6, and 7 are confounded respectively with the interactions 12, 23, 123, and 13.
Interpretation of results
The plans of Plackett and Burman pose the same problem of interpretation of the results as the fractional planes. As we neglect the interactions, we must take into account the theory of aliases for the interpretation of the results.
1 – J. J. Droesbeke, J. Fine, G. Saporta (1997) – Experience Plans – Business Applications
2 – D. Behera, Y. Tourbier, S. Germain-Tourbier (1994) – Plans of experiments: Construction and analysis
R. L. Plackett, J. P. Burman (1946) – The design of Optimum multifactorial experiments